Conformal field theory in two dimensions/Exactly solvable theories

Spectrums and loop CFTs

See Sections 2.1 and 2.4 of Ref.

Exercises

• ES1. In the Ising minimal model, what are the OPE spectrums? Write the channel spectrums of all inequivalent 4-point functions.
• ES2. In a generalized minimal model, write the 3 channel spectrums of $\left\langle V_{\langle 3,3\rangle }^{d}V_{\langle 2,4\rangle }^{d}V_{\langle 1,4\rangle }^{d}V_{\langle 2,5\rangle }^{d}\right\rangle$.
• ES3. What are the largest possible non-diagonal spectrums for a CFT with 2 degenerate fields, such that all spins are integer?

ESDI: Conformal dimensions of non-diagonal fields

We consider the non-diagonal primary fields with integer conformal spins, in a CFT that has a degenerate field $V_{\langle 1,3\rangle }^{d}$. We assume $\Re \beta ^{2}>0$.

1. Which non-diagonal fields $V_{(r,s)}$ can exist?
2. Show that the set of their total conformal dimensions $\Delta +{\bar {\Delta }}$ is discrete and bounded from below.
3. Show that there are finitely many fields whose total dimension is below any given bound.
4. Assuming $\beta ^{2}\in \mathbb {R}$, sketch the set of non-diagonal fields on a plot whose axes are $\Delta -{\bar {\Delta }}$ and $\Delta +{\bar {\Delta }}$.

Liouville theory and non-diagonal minimal models

Liouville theory

See Section 2.2.2 of Ref.

D-series minimal models

See Section 2.3.1 of Ref.

Exercises

• EL1. From OPEs in generalized minimal models, deduce the OPEs in Liouville theory with $c\leq 1$. In particular, show that the measure of integration over momentums is the uniform measure $dP$. (You may use a Fourier transform.)
• EL2. What is the intersection of the spectrums of the A-series and D-series minimal models at the same central charge? Is this the spectrum of a consistent CFT?
• EL3. Show that the spectrum of the D-series minimal model DMM$_{6,5}$ is a subset of the spectrum of the 3-state Potts CFT. Explain why the diagonal primary field $V_{P_{(0,{\frac {1}{2}})}}$ appears in the non-diagonal sector of DMM$_{6,5}$. Admitting that $V_{P_{(0,{\frac {1}{2}})}}$ transforms in the standard representation of the symmetric group $S_{3}$, explain why it appears twice in DMM$_{6,5}$.

ESES: E-series minimal models

1. Find the spectrums of E-series minimal models in Wikipedia.
2. Which diagonal degenerate fields exist in E-series minimal models?
3. Compare the spectrums of the A-series, D-series and E-series minimal models with the same central charge.
4. In the case $q=12$, rewrite the spectrum in terms of non-diagonal fields $V_{(r,s)}$, and compare with loop models.